Polish groupoids and functorial complexity

TL;DR

This paper introduces a functorial Borel complexity for Polish groupoids, linking it to the classification complexity of objects and extending descriptive set theory results to groupoid actions.

Contribution

It defines functorial Borel complexity for Polish groupoids and generalizes key descriptive set theory results to groupoid actions, answering a notable open question.

Findings

Functorial Borel complexity coincides with Borel complexity for treeable orbit relations.

Existence of Polish groupoids with different complexities sharing the same orbit relation.

Generalization of fundamental results about Polish group actions to groupoid actions.

Abstract

We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids with essentially treeable orbit equivalence relation, functorial Borel complexity coincides with the Borel complexity of the associated orbit equivalence relation. On the other hand for every countable equivalence relation $E$ that is not treeable there are Polish groupoids with different functorial Borel complexity both having $E$ as orbit equivalence relation. In order to obtain such a conclusion we generalize some fundamental results about the descriptive set theory of Polish group actions to actions of Polish groupoids, answering a question of Arlan Ramsay. These include the Becker-Kechris results on Polishability of Borel $% G $-spaces, existence of universal Borel $G$-spaces, and characterization of Borel $G$-spaces with Borel orbit equivalence relations.